# determinant

• November 30th 2010, 01:56 PM
alexandrabel90
determinant
given that A is an nxn matrix, how do you show that the characteristic polynomial of A is (-1)^n det(A)?
• November 30th 2010, 03:54 PM
tonio
Quote:

Originally Posted by alexandrabel90
given that A is an nxn matrix, how do you show that the characteristic polynomial of A is (-1)^n det(A)?

Doing magic because the above is far from being true. In fact, $(-1)^n\det A$ is not even a polynomial of degree n, let alone

the characateristic polynomial of any nxn matrix. Read carefully again the question and repost.

Tonio

PS Perhaps there is something about some free coefficient or stuff...?(Giggle)
• November 30th 2010, 05:47 PM
alexandrabel90
the exact question is let A be an nxn matric and the trace of A is defined by tr(A)= a_11+a_22...+a_nn . Prove that the constant term of the characteristic polynomial is (-1)^n det(A)
• November 30th 2010, 06:12 PM
Drexel28
Quote:

Originally Posted by alexandrabel90
the exact question is let A be an nxn matric and the trace of A is defined by tr(A)= a_11+a_22...+a_nn . Prove that the constant term of the characteristic polynomial is (-1)^n det(A)

You know that $p(x)=\det\left(Ix-A\right)$, right? And that the constant term of a polynomial $q$ is $q(0)$ and $\det\left(c A\right)=c^n\det(A)$...right?
• November 30th 2010, 06:27 PM
alexandrabel90
yup, i know that..

so lets say, A is similar to an upper triangular matric so P(x) = (x-a_11)....(x-a_nn) = x^n + a_n-1 x^(n-1) +....+a_0 so P(0) = a_0 and then how do i continue?
• November 30th 2010, 06:37 PM
Drexel28
Quote:

Originally Posted by alexandrabel90
yup, i know that..

so lets say, A is similar to an upper triangular matric so P(x) = (x-a_11)....(x-a_nn) = x^n + a_n-1 x^(n-1) +....+a_0 so P(0) = a_0 and then how do i continue?

Friend, what if we just plugged $0$ into $p(x)=\det\left(Ix-A\right)$?
• November 30th 2010, 06:40 PM
alexandrabel90
oh crap!!! im so blur! thanks